Question

Rainbow is an arch of colours that is visible in the sky after rain or when water droplets are present in the atmosphere. The colours of the rainbow are generally red, orange, yellow, green, blue, indigo, and violet. Each colour of the rainbow makes a parabola. We know that any quadratic polynomials p(x) = ax2 + bx + c (a ≠ 0) represents a parabola on the graph paper.

Based on the above information, answer the following questions.

1. The graph of a rainbow y = f(x) is shown in the figure given below. Write the number of zeros of the curve.
   
2. If the graph of a rainbow does not intersect the x-axis but intersects y-axis at one point, then how many zeros will it have? 
3. The polynomial x² – 2x – (7a + 3) represents a rainbow. If –4 is a zero of it, find the value of a. 
4. If α and β are the zeros of the quadratic polynomialp (x) = x² – rx – s then find the value of α² + β².

Answer 1

i. The downward-facing parabola cuts the x-axis at two points. Each point of intersection with the x-axis is a zero, so the curve has 2 zeros.

ii. A zero is an x-coordinate where the graph meets the x-axis.

Meeting the y-axis does not create a zero.

Since the graph never touches the x-axis, the polynomial has no zeros.

iii. Given: zero of the polynomial is x = –4.

Substitute x = –4:

\[(-4)^{2}-2(-4)-(7a+3)=0\]

\[16+8-7a-3=0\]

\[21-7a=0\]

\[a=3\]

iv. For p(x) = x² – rx – s:

\[\alpha+\beta=r,\quad \alpha\beta=-s\]

\[\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta\]

\[\alpha^{2}+\beta^{2}=r^{2}-2(-s)=r^{2}+2s\]